VOLUME 89, NUMBER 6
PHYSICAL REVIEW LETTERS
5 AUGUST 2002
Vortices with Fractional Flux in Two-Gap Superconductors and in Extended Faddeev Model
Egor Babaev
Institute for Theoretical Physics, Uppsala University, Box 803, 75108 Uppsala, Sweden
NORDITA, Blegdamsvej 17, DK-2100 Copenhagen, Denmark
Science Institute, University of Iceland, Dunhaga 3, 107 Reykjavik, Iceland
(Received 17 November 2001; published 23 July 2002)
We discuss linear topological defects allowed in two-gap superconductors and equivalent extended
Faddeev model. We show that, in these systems, there exist vortices which carry an arbitrary fraction of
magnetic flux quantum. Besides that, we discuss topological defects which do not carry magnetic flux and
describe features of ordinary one-magnetic-flux-quantum vortices in the two-gap system. The results
could be relevant for the newly discovered two-band superconductor MgB2 .
DOI: 10.1103/PhysRevLett.89.067001
PACS numbers: 74.60.–w, 11.27.+d, 74.76.Db
A fundamental property of the Abelian Higgs model is
the quantization of magnetic flux [1]. In an ordinary superconductor the Abrikosov vortices can carry only an integer
number of magnetic flux quanta. The intriguing possibilities of topological defects carrying a fraction of flux
quantum have long attracted interest, and several nontrivial
realizations were identified. For example, a half fluxquantum vortex in a spin-1 condensate is a configuration
where a Cooper pair has its spin reversed when moving
around the vortex core (this is an analog of an Alice string
in high energy physics where a particle moving around the
string flips its charge or enters a ‘‘shadow world’’); also, a
half flux-quantum vortex can be formed on a junction of
3-grain boundaries in a crystal, etc. [2]. In this Letter, we
discuss vortices in two-gap superconductors [3,4] (known
in particle physics as a Higgs doublet model [5]) and in the
extended Faddeev model. We show that these vortices can
carry an arbitrary fraction of magnetic flux quantum.
Experimentally, two-gap superconductivity has been
observed in the transition metals Nb, Ta, V, and in
Nb-doped SrTiO3 [6]. More recent experiments indicate
the two-gap nature of superconductivity in MgB2 [7] and
2H NbSe2 [8]. Two-gap models appear also in the theoretical studies of liquid metallic hydrogen, which should
allow superconductivity of both electronic and protonic
Cooper pairs [9]. Other realizations of the two-gap system
are superconductors with two types of pairing (e.g., a
mixture of s- and p-wave condensates).
A two-gap superconductor is described by a two-flavor
Ginzburg-Landau (GL) free energy functional:
F
1
1
jr ieA1 j2 jr ieA2 j2
2m1
2m2
B2
;
Vj1;2 j2 1 2 2 1 2
(1)
where j
jei
, Vj1;2 j2 b
j
j2 c
4
2 j
j , and is a characteristic of the interband
Josephson coupling strength [4]. Many exotic properties
of (1) are obscured in the GL presentation of the free
energy functional. In [10] it was shown that there exists
067001-1
0031-9007=02=89(6)=067001(4)$20.00
an exact equivalence mapping between the model (1) and
an extended version of Faddeev’s O3 nonlinear model
[11], which describes the two-gap superconductors in
terms of gauge invariant variables which explicitly show
the degrees of freedom present in the system. This model
consists of a three-component unit vector n~ in interaction
~ and a density-related variable
with a massive vector field C
[12]:
F
2
2 ~ 2
rn~ 2 r2 C
V 2 Kn1
4
16
1
@i Cj @j Ci n~ @i n~ @j n~ 2 ;
32e2
(2)
where @i dxd i , V A Bn3 Cn23 . The models (1) and
(2) are connected in the following way [10]: coefficients A; B; C are given by A 2 4c1 m21 4c2 m22 b1 m1 b2 m2 ; B 2 8c2 m22 8c1 m21 b2 m2 b1 m1 ;
C 42 c1 m21 c2 m22 . The position of the unit vector
n~ on the sphere S2 can be characterized by two angles as
follows: n~ sin cos;
sin
p
p where ~ sin; cos,
1 2 ; j1;2 j 2m1 sin2; 2m2 cos2
; C
i
i
f
r
r
g
f
r
r
g
2
2
1
1
2
2
1
1
2
2 m m 1
2e j1 j2
2 m1
2
2
jm22j A. We consider the system in the presence
of an interband Josephson coupling 1 2 2 1 p
2 Kn1 , where K 2 m1 m2 . The potential term V in (2)
determines the ground state value for n3 , which corresponds to uniform density of both condensates. We denote
N2
N1 N1
N2 1
~ 3 m
it as n
m
m1 m
, where N1;2 stands for the
2
1
2
2
average hj1;2 j i. In the presence of the intrinsic
Josephson effect, the ground state value of n~ corresponds
to a point where n1 1 on a circle specified by the
~ . We note that (2) shows the
~ 3 cos
condition n3 n
~ ),
Meissner effect [10] (the generation of mass for C
the corresponding length scale is the London magnetic
2
2
field penetration length: 2 e12 jm11j jm22j 1 . One
can observe from (2) that along with the Meissner effect
in a two-band superconductor there exists a neutral boson.
That is, if one considers a uniform density of
~ decouples from the field n~ ,
condensates, the vector field C
 2002 The American Physical Society
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VOLUME 89, NUMBER 6
PHYSICAL REVIEW LETTERS
because n~ @i n~ @j n~ / sin@i @j @i @j 0
when n3 cos~ const. Thus, when 0, the system
possesses a massless neutral O2 excitation associated
with the phase variable 1 2 . This phenomenon
has no counterpart in one-gap superconductors where the
Goldstone boson associated with O2 symmetry is
5 AUGUST 2002
‘‘eaten’’ by the Meissner effect. In the presence of the
Josephson effect ( 0) the variables still decouple, but
the Josephson term 2 Kn1 breaks the neutral O2 symmetry by giving a mass to n1 .
Let us discuss vortices in the system (2). In the London
limit (ji j const) , Eqs. (1) and (2) become
2
2 ~ 2
1
rn~ 2 C
@i Cj @j Ci 2 2 Kn1
4
16
32e2
2
~ ~ 2
B2
~ cos:
2 K sin
r1 cos2
r2 eA sin2 ~ r2 2 sin2
2
2
4
2
F
From this expression, one can observe
that the vortices
H
characterized by 1 2 dl r1 2 4#m; 1 2 0 (where we integrate over a closed
curve around the vortex core) are the analog of m-flux
quanta Abrikosov vortices in an ordinary superconductor
2
j1 j2
j2 j2
characterized by jj
m m1 m2 . Let us observe that if
both phases 1;2 change by 2# around the core, then a
vortex carries one quantum of magnetic flux.
The vortices characterized by 1 2 4#n in the
2
2
case where jm11j jm22j have nontrivial structure. Let us
first consider the case when 1 2 0 and 1 2 4#. First of all, such a vortex features a neutral
superflow characterized by a 4# gain in the variable . In
the case of 0, this vortex is described by the
~ r2 sine-Gordon functional:
F 1=42 sin2 2
~
K sin cos. In the case K 0, this vortex is
equivalent to a vortex in a neutral superfluid with super~ . In the case when j1 j2 j2 j2 ,
fluid stiffness 1=22 sin2 m1
m2
for a topological defect where 1 2 0 and
1 2 4#, the second term in (3) becomes
~
~
2 fsin2 2 cos2 2 g r122 eA
2 . This term is nonvanishing for such a vortex configuration, which means
that this vortex besides neutral vorticity also carries magnetic field. Let us calculate the magnetic flux carried by
such a vortex. The supercurrent around the core of this
~
~
vortex is J 2e2 fsin2 2 cos2 2 g r122 eA
. Let
us now integrate this expression over a closed path situated at a distance larger than from the vortex core.
Indeed, at a distance much larger than the penetration
length the supercurrent J, or equivalently the massive
~ , vanishes. Thus we arrive at the following equation:
field C
I 1
~ 0 n
~ 3 0 ; (4)
r1 2 dl cos
2
H
where A dl is the magnetic flux carried by our
vortex, and 0 2#=e is the standard flux quantum. From
(4) it follows that, in our system, such a vortex can carry an
arbitrary fraction of magnetic flux quantum since it de~ , which is a measure of
pends on the free parameter cos
the relative densities of the two condensates in the system.
In the general case, a vortex characterized by the following
cos~
067001-2
(3)
phase changes around the core 1 2#k1 and 2 2#k2 carries the following flux:
"
!
! #
~
~
2 2 k cos
k 0 :
(5)
k1 ;k2 sin
2 1
2 2
We should emphasize that a straightforward inspection
of (2) shows that this model allows neutral vortices associated with the neutral O2 boson without a nontrivial
~ for any values of n
~ 3 . However,
configuration of the field C
~ 3 0 charactersuch solutions (e.g., vortices in the case n
~ 0) are unphysical because these
ized by 2#; C
vortices do not satisfy the condition that i change by 2#ki
around the vortex core [as follows from (3)]. Thus, while
the original model (1) and the extended Faddeev model (2)
have the same number of degrees of freedom so that the
~ are dynamically independent by construcfields n~ , C
tion, the mapping incurs a constraint on topological defects
~ since 1;2 appear in both of them. So, in the
in n~ and C
hydrodynamic limit in (2), there is no direct coupling
~ ; however, in a non-simply
between the fields n~ and C
connected space a topological defect in the field n~ neces~ . The
sarily induces a nontrivial configuration of the field C
consequence of this is the fractionalization of magnetic
flux in the model (2).
So the models (1) and (2) allow the following linear
composite topological defects:
(i) 1 2 4#n; 1 2 0: These are
the vortices which feature neutral superflow. When the
ground state of the variable n~ corresponds to the equator
2
2
on S2 (that is, the case when jm11j jm22j or, equivalently,
cos~ 0), these vortices do not carry magnetic flux.
~ 0, the only physical solutions
In the case when cos
with neutral vorticity also carry a fraction of magnetic flux
quantum. In the case of nonzero Josephson coupling, these
vortices are described by the sine-Gordon equation (3).
(ii) 0; 1 2 4#k: These are the vortices which feature circular supercurrent and no circular
neutral superflow. Such vortices carry k flux quanta of
magnetic field. One of the physical consequences of
our analysis is that we can observe that, in the two-gap
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VOLUME 89, NUMBER 6
PHYSICAL REVIEW LETTERS
superconductor (1) and (2), in spite of the existence of two
types of Cooper pairs, a formation of two sublattices of
vortices corresponding to each of the condensates in an
external field is energetically forbidden. This is because the
energy per unit length of noncomposite vortices is divergent in an infinite sample in cases of both zero and nonzero
Josephson coupling (in case of finite , a vortex creates a
domain wall which makes its energy per unit length
divergent in an infinite sample [13,14]). We can also
observe that a superconductor made up of one condensate
which is of type-II and another condensate which is of
type-I will, in general, in the presense of an external
magnetic field, form one-flux-quantum vortices involving
both condensates and will preserve two-gap superconductivity, even if the external field exceeds the thermodynamic
critical magnetic field for the type-I condensate.
(iii) 2#n; 1 2 4#l: These are the
vortices which may be viewed as a cocentered l-flux
quanta Abrikosov vortex combined with a vortex with
neutral vorticity with winding number n carrying a fractional magnetic flux. These vortices and the vortices of
type (ii) characterized by 1 2 8# could be
unstable against decay into more simple vortices [13].
Besides composite vortices, the system allows vortices
where the phase of one of the condensates changes by
2# around the core, whereas the phase of the second
condensate remains constant. These vortices are also principally different from ordinary Abrikosov vortices. Let us
discuss these vortices in the GL formalism (1). This will
provide additional perspective on the discussion and will
allow us to illustrate the connection between the models (1)
and (2). We shall present a solution for these vortices in the
limit when the inverse mass for n1 , which is given by the
Josephson term, is larger than the sample size. Such an
approximation allows one to put to zero. We should note
that there certainly exist systems where is exactly zero.
That is, a model with 0 should describe vortices in a
bilayer system: a superconductor-insulator-superconductor
compound is an example of a system of two condensates
coupled only by a gauge field. In such a system, a vortex in
one layer carries flux through the second layer, where it
also induces a current, which should lead to the flux
fractionalization. Besides that, an additional U1 symmetry appears if one considers equal-spin pairing. In that
case, if the spin-orbit coupling is neglected, each spin
population has its own phase, without Josephson cou
pling [15].
Let us recall the GL equation for the supercurrent in
ie
ie
standard notations: J 2m
1 r1 1 r1 2m
1
2
2
2
2 r2 2 r2 e2 Ajm11j jm22j . When only the
phase 1 changes by 2# around the core while the phase of
the second condensate remains constant, we have the
following expression for the vector potential:
2
2
2
2
2
A Je2 jm11j jm22j 1 1e jm11j jm11j jm22j 1 r’1 .
From this expression, it follows that the vortex character067001-3
5 AUGUST 2002
ized by 1 2#; 2 0 carries the following fractional magnetic flux:
I
2
2
2 1
~ dl j1 j j1 j j2 j
0 : (6)
A
m1
m1
m2
Let us now remark on what are the physical roots of this
flux fractionalization. Further examining the solution for
the vortex (1 2#, 2 0), we can write the vector
z
potential as A re
jrj jArj, where r measures the distance
from the core and ez is a unit vector pointing along the
d
core. The magnetic field is then given by jBj 1r dr
rjAj.
The equation for the current J can then be rewritten as
d 1d
j j2
e
rjAj 1 jAje2 dr r dr
r
m1
j2 j2
jAje2 0:
m2
For such a vortex, the solution for the vector potential is
jAj j1 j2
m1
j1 j2
j2 j2
m1 m2
j1 j2
m1
1
er
K1
q
j1 j2
j2 j2
m1 m2
s !
j1 j2 j2 j2
e
r :
m1
m2
(7)
Indeed, the magnetic field vanishes exponentially away
from the vortex core with the characteristic length scale
given qby
the magnetic
field penetration
length
q
2
2
2
2
2
e jm11j jm22j 1 : jBj e jm11j K0 e jm11j jm22j r.
In contrast to the Abrikosov vortex [1], besides having
the fractionalization of magnetic flux, our vortex also
features the neutral vorticity. This, in particular, can be
seen by substituting the solution (7) into (1). Then, at
length scales larger than from the vortex core, we have
the following expression for the energy density: F 2
j1 j2 j1 j2
1
jm22j 1=2 1r1 j2 2m1 2 j 2m1 jr i m1 m1
2
2
2
i jm11j jm11j jm22j 1=2 1r2 j2 . Thus the energy per unit
length of the vortex (1 2#, 2 0) is divergent.
This is due to the fact that such a topological configuration
induces in a two-gap system the neutral superflow. Indeed,
the above expression for F is similar to the energy density
of a vortex in a neutral system with effective stiffness
j1 j2 j2 j2 j1 j2
j2 j2 1
m1 m2 m1 m2 :
1 j1 j2 j2 j2 j1 j2 j2 j2 1
Fn jrei1 j2 : (8)
2 m1
m2
m1
m2
This shows transparently the physical origin of the presence in the system of a massless neutral boson; a topologically nontrivial configuration (1 2#, 2 0),
besides having a current in the condensate 1 , also necessarily induces current in the condensate 2 . Albeit in such
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VOLUME 89, NUMBER 6
PHYSICAL REVIEW LETTERS
a configuration there are no gradients of 2 ; however, the
two condensates are not independent but are connected by
the vector potential. The admixture of an oppositely directed supercurrent of the condensate 2 leads to the
situation when the two supercurrents partially compensate
the magnetic field induced by each other. This leads to the
existence of the effective neutral superflow in the system.
Moreover, it is the fact that the two currents partially
compensate the magnetic field induced by each other
which leads to the fractionalization of magnetic flux. We
stress that the vortex solutions (1 2#, 2 0) in
this model, albeit being topologically stable, cannot form
as an energetically preferred state in external field. In an
external field the system will form the composite vortices
(1 2#, 2 2#) described above.
There is, however, an experimental setup which should
allow one to directly observe (1 2#, 2 0) excitations. That is, in a two-gap superconducting film, there
should occur thermal creation of pairs of vortices and
antivortices with neutral vorticity and fractional magnetic
flux. The fact that this type of vortex features both confined
magnetic field and long-range interaction due to neutral
superflow will allow (i) detection of the BerezinskiiKosterlitz-Thouless (BKT) transition in a system of these
vortices by standard experimental techniques such as fluxnoise measurements or measurements in an applied dynamic magnetic field (such as in a system of Abrikosov
vortices [16]). (ii) Such measurements allow one to extract
data about the type of interaction between a vortex and an
antivortex in the system. In spite of being by definition
sensitive only to magnetic flux carrying vortices, these
measurements should give the picture of the BKT transition such as in a neutral U1 system. That is, the transition
should be a pure BKT transition but not a ‘‘would-be’’
BKT transition such as in a system of Abrikosov vortices.
As is well known, there is no true BKT transition in a
system of Abrikosov vortices: because their interaction is
screened by the Meissner effect and a BKT transition is
replaced by a crossover which is observable only when
penetration length is sufficiently large [17]. Moreover, the
BKT transition in a system of these vortices should be
observable even in a type-I system and in both limits 0 and when is large, where one has sine-Gordon vortices
interacting with a linear potential [13].
In conclusion, we have shown that a two-gap superconductor allows several types of vortices, such as vortices
carrying an arbitrary fraction of the magnetic flux quantum, which have no counterpart in ordinary one-gap superconductors. This study also has interdisciplinary interest
because similar models are highly relevant in particle
physics [5,18].
067001-4
5 AUGUST 2002
It is a great pleasure to thank L. D. Faddeev, D.
Gorokhov, A. J. Niemi, G. Volovik, K. Zarembo, V.
Cheianov, O. Festin, S. Girvin, and P. Svedlindh for discussions and/or comments. This work has been supported
by Grant No. STINT IG2001-062, the Swedish Royal
Academy of Science, Göran Gustafsson Stiftelse UU/
KTH, and by NorFA.
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[2] For a recent review and citations, see G. E. Volovik, Proc.
Natl. Acad. Sci. U.S.A. 97, 2431 (2000).
[3] H. Suhl, B. T. Matthias, and L. R. Walker, Phys. Rev. Lett.
3, 552 (1959).
[4] A. J. Leggett, Prog. Theor. Phys. 36, 901 (1966).
[5] T. D. Lee, Phys. Rev. D 8, 1226 (1973).
[6] See, e.g., L. Shen et al., Phys. Rev. Lett. 14, 1025 (1965);
G. Binnig et al., Phys. Rev. Lett. 45, 1352 (1980).
[7] See, e.g., F. Bouquet et al., Phys. Rev. Lett. 87, 047001
(2001); Amy Y. Liu et al., Phys. Rev. Lett. 87, 087005
(2001); P. Szabo et al., Phys. Rev. Lett. 87, 137005 (2001).
[8] T. Yokoya et al., Science 294, 2518 (2001).
[9] K. Moulopoulos and N. W. Ashcroft, Phys. Rev. B 59,
12 309 (1999); J. Oliva and N. W. Ashcroft, Phys. Rev. B
30, 1326 (1984); N. W. Ashcroft, J. Phys. A 129, 12
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65, 100512 (2002).
[11] L. Faddeev, IAS Report No. Print-75-QS70, 1975; in
Relativity, Quanta and Cosmology, edited by M. Pantaleo and F. De Finis, Vol. 1 (Johnson Reprint, New
York, 1979).
[12] A similar model has been found to be relevant for spintriplet superconductors: E. Babaev, Phys. Rev. Lett. 88,
177002 (2002).
[13] More details will be presented elsewhere. See also E.
Babaev cond-mat/0201547. E. Babaev and D. Gorokhov
(to be published).
[14] The energy of the fractional flux vortices in such a system
will, however, be finite in a mesoscopic sample. Also, in a
granular superconductor, lattice effects may raise the mass
of the neutral O2 boson, which will make the energy per
unit length of this defect finite.
[15] The author thanks G. Volovik for pointing this out.
[16] P. Minnhagen, Rev. Mod. Phys. 59, 1001 (1987).
[17] The same applies to the Pearl vortices: J. Pearl, Appl.
Phys. Lett. 5, 65 (1964); in Low Temperature Physics-LT
9, edited by J. G. Daunt et al. (Plenum, New York, 1965),
p. 566.
[18] L. Faddeev and A. J. Niemi, Nature (London) 387, 58
(1997); Phys. Rev. Lett. 82, 1624 (1999); Phys. Lett. B
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Phys. Rev. D 65, 025005 (2002); Y. M. Cho, H. W. Lee,
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